Cognitive Computing: 2012

1. Cognitive Computing: 2012 Consciousness and Computation: computing machinery & intelligence 3. EMULATION Mark Bishop

2. The Universal Machine Norma • Thesis: “With suitable coding of data, every algorithm can be represented as a flowchart program for NORMA.” • But how? Some obvious criticisms of NORMA include... • 1: NORMA does not define enough operations and tests: • No assignment, multiply, subtract, divide etc. • 2: NORMA data types are too restricted: • No provision for negative or floating point numbers. • 3: Access to data is too restricted: • No provision for ARRAYs of numbers. • 4: The restriction to flowchart programs is too restrictive: • No mechanism for procedures. At the very least one needs to be able to refer to labels indirectly so that sub-routines can be constructed. • 5: Other criticisms include: • The lack of string handling. • No logical shifts etc. • The criticisms [1..4] are considered the most important. (c) Bishop: Consciousness and Computations

3. Informal ‘proof’ of thesis • To demonstrate that every algorithm can be coded as a NORMA flowchart program it is necessary to show how each of the main criticisms [1..4] can be answered. • Specify a new machine NORMA+ which has the desired feature. • Show how any program in NORMA+ can be –recoded as a flowchart program for NORMA. (c) Bishop: Consciousness and Computations

4. Answer to 1st criticism • Assignment to a constant (eg. A := 0) • Theorem: “Every WHILE program can be mapped onto a corresponding flowchart program”. WHILE (A <> 0) DO A := A -1; END; This WHILE program can be written as a ‘macro’. Similar macros can be written for (A := 1), (A := 2) etc. (c) Bishop: Consciousness and Computations

5. Addition A := A + B using [a]; Consider a macro to calculate (A := A + B). We need to use an extra register a, hence we write the macro as {A := A + B using a} a := 0; WHILE (B <> 0) DO {adds B into A and makes a copy of B} A := A + 1; a := a + 1; B := B - 1; END; WHILE (a <> 0) DO {resets B to original value} a := a - 1; B := B + 1; END; (c) Bishop: Consciousness and Computations

6. Assignment to a register A := B using [a]; A := 0; A := A + B using a; (c) Bishop: Consciousness and Computations

7. Subtraction A := A - B using [a, b]; a := B using b; WHILE (a <> 0) DO A := A - 1; a := a - 1; END; (c) Bishop: Consciousness and Computations

8. Register Test Operations 1 ?(A < 2) using [a, b]; IF ?(A < 2) THEN L2: TRUE ELSE L1: FALSE END; a := A using b; IF (a = 0) THEN GOTO L2; a := a - 1; IF (a = 0) THEN GOTO L2; L1: FALSE; L2: TRUE; (c) Bishop: Consciousness and Computations

9. Multiplication A := A x B using a, b, c; a := A using b; b := B using c; A := 0; WHILE (b <> 0) DO A := A + a using c; b := b - 1; END; (c) Bishop: Consciousness and Computations

10. Division A := A DIV B using [a..e]; {Remainder held in e} e := A using a; A := 0; WHILE (e >= B using [a..c]) DO e := e - B using d; {[a..c] still in use} INC A END; (c) Bishop: Consciousness and Computations

11. NORMA and primes: on testing for integer division Div (A, B)? {True iff A is exactly divisible by B} a := A; WHILE (a >= B) DO a := a – B; IF (a = 0) THEN GOTO Label-Divisible ELSE GOTO Label-NOT-Divisible (c) Bishop: Consciousness and Computations

12. On the testing for prime aPrime (A)? {NB. Zero and one are neither prime nor composite} IF (A < 2) THEN GOTO [Label-NOT-Prime] j := A - 1; WHILE NOT (Div (A, j) DO j := j – 1; IF (j = 1) THEN GOTO [Label-Prime] ELSE GOTO [Label-NOT-Prime] (c) Bishop: Consciousness and Computations

13. On calculating the kth prime A := PRIME (K) {Stores the Kth prime in A, where the 1st prime is 2} A := 0; k := K; WHILE (k <> 0) DO k := k – 1; A := A + 1; WHILE NOT ( aPrime (A) ) DO A := A + 1; END; (c) Bishop: Consciousness and Computations

14. Answer to criticism 2 Representing negative numbers An arbitrary integer m can easily be represented as the order pair (n, d) of non negative integers: n = |m| d = 0 IF (m >= 0) d = 1 otherwise (c) Bishop: Consciousness and Computations

15. Representing fixed length real numbers • Using a similar idea to that used for negative numbers, operations on a non negative rational number r can be defined in terms of the ordered pair (a, b), where (b > 0) and (r = a/b). • Since arithmetic on rationals conforms to the following rules: • Addition & Subtraction • (a,b) ± (c,d) = (ad ± bc, bd) • Multiplication • (a,b) × (c,d) = (ac, bd) • Division • (a,b) / (c,d) = (ad, bc) iff (c <> 0) • Equality • ?((a,b) = (c,d)) iff (ad = bc) • ... there is no problem constructing appropriate NORMA programs using fixed length reals. (c) Bishop: Consciousness and Computations

16. Answer to 3rd criticism • To answer criticism 3 we define a new machine SAM (Simple Array Machine) with more flexible access to data. • SAM augments NORMA by possessing the array of registers, A[1], A [2] .... A [n] • in addition to the standard registers A,B .... Y, which are now referred to as index registers. • The operations defined by SAM are those of NORMA plus the array operations: • A [J] := A [J] + 1; • A [J] := A [J] - 1; where J is any index register. • A [n] := A [n] + 1; • A [n] := A [n] - 1; where n is any positive integer. (c) Bishop: Consciousness and Computations

17. Array test operations • SAM also has the array test operations: ?(A [J] = 0) ?(A [n] = 0) • SAMs input and output functions are the same as NORMAs except that the input function also initialises each array register to zero. (c) Bishop: Consciousness and Computations

18. Theorem 2: NORMA can simulate SAM • Proof: We have to show how any given program P for SAM can be translated into a NORMA program Q such that: • NORMA (Q) = SAM (P) • Method: Pack all of SAM array into a single NORMA register. • If at some stage in the computation a SAM array contains [a1, a2, .. an] then at the equivalent stage a NORMA register will contain A. • A = P1a1× P2a2× P3a3 ... × Pnan, where Pj = jth prime. (c) Bishop: Consciousness and Computations

19. Increment indexed array • To define Q from P, we translate each SAM instruction into a sequence of NORMA instructions as follows: • All index register instructions are left unchanged. • Array operations of the form A [J] := A [J] + 1 are translated into: • B := PRIME (J); • A := A × B • Where PRIME is the PRIME number function defined earlier. (c) Bishop: Consciousness and Computations

20. Decrement indexed array • Array operations of the form A [J] := A [J] - 1 are translated into: B := PRIME (J); A := A DIV B; Where DIV is a special integer division defined by: a DIV b = a / b If b divides exactly into a = a otherwise (c) Bishop: Consciousness and Computations

21. Testing an array element • A test of the form ?(A [J] <> 0) is translated into: B := PRIME (J) RETURN div (A, B) Where div (A ,B) is TRUE when A is exactly divisible by B and FALSE otherwise. ie. The test div (A, B) will return TRUE just in the case that A [J] ≠ 0. (c) Bishop: Consciousness and Computations

22. INC and DEC array using a constant index, n • An operation of the form (A [n] = A [n] + 1) is translated into: B := PRIME (n); A := A × B; • Similarly (A [n] = A [n] - 1) is translated into: B := PRIME (n); A := A DIV B; (c) Bishop: Consciousness and Computations

23. Proof of theorem 2 • The test ?(A [n] = 0) uses the test div (Pn, A) to RETURN the correct value, where Pn = PRIME (n). • Now each operation and test of Q must be replaced by the corresponding NORMA macro; • And if we ensure that Q initially sets A to 1 to represent the input condition of SAMs array then … • … the simulation clearly works and hence Theorem 2 is proved. (c) Bishop: Consciousness and Computations

24. Answer to criticism 4 - only flowcharts • To answer criticism 4 we need to define a new machine SIM (Simple Indirect Machine). • In a SIM program P with labels (1 .. l .. k), the operations defined by SIM are those given by NORMA plus the following two Indirect Jump calls: l: GOTO (A); {GOTO a where a is the content of register A} l: IF (T) THEN GOTO (A); {IF TRUE GOTO label a} (c) Bishop: Consciousness and Computations

25. Theorem 3: NORMA can simulate SIM • Proof: We have to show how any given program P for SIM can be translated into a NORMA program Q such that: • NORMA (Q) = SIM (P) • Method: Use a Jump Table. • For each register (eg. A) that appears in the program we need to define an extra segment of code, tableA. (c) Bishop: Consciousness and Computations

26. Jump tables (c) Bishop: Consciousness and Computations

27. For the SIM Indirected jump • This extra code can be added after instruction k of SIM program P. • We can now replace the new SIM instructions by the following NORMA code: • SIM • l: GOTO (A); • NORMA • l: GOTO tableA; {Where tableA is label of jump table for the A reg} (c) Bishop: Consciousness and Computations

28. The SIM indirected test • SIM • l: IF (T) THEN GOTO (A); • NORMA • l: IF (T) THEN GOTO tableA; {Where tableA is label of • the jump table for the A register} • Now each operation and test of SIM Q must be replaced by the corresponding NORMA macro. • It is now clear that the resulting NORMA program Q is equivalent to the SIM program P. • The simulation clearly works and hence Theorem 3 is proved. (c) Bishop: Consciousness and Computations

29. HOMEWORK:Implement NORMA_STACK • Prove that the machine NORMA_STACK is no more powerful than the universal machine NORMA by designing two MACROs to implement: • (a) X=POP which removes the top value from the STACK and places it into the X register; • (b) PUSH (X) which places the contents of the X register on to the top of the STACK; • and submit a short (no more than 1-page) report detailing their operation. (c) Bishop: Consciousness and Computations